The ancient Andean empire built great cities but left no written records – except perhaps in mysterious knotted strings called khipu. Can an anthropologist and some mathematicians crack the code?

Incan civilization was a technological marvel. When the Spanish conquistadors arrived in 1532, they found an empire that spanned nearly 3,000 miles, from present-day Ecuador to Chile, all served by a high-altitude road system that included 200-foot suspension bridges built of woven reeds. It was the Inca who constructed Machu Picchu, a cloud city terraced into a precarious stretch of earth hanging between two Andean peaks. They even put together a kind of Bronze Age Internet, a system of messenger posts along the major roads. In one day, Incan runners amped on coca leaves could relay news some 150 miles down the network.

Yet, if centuries of scholarship are to be believed, the Inca, whose rule began 2,000 years after Homer, never figured out how to write. It’s an enigma known as the Inca paradox, and for nearly 500 years it has stood as one of the great historical puzzles of the Americas. But now a Harvard anthropologist named Gary Urton may be close to untangling the mystery.

His quest revolves around strange, once-colorful bundles of knotted strings called khipu (pronounced KEY-poo). The Spanish invaders noticed the khipu soon after arriving but never understood their significance – or how they worked.

Once, at the beginning of the 17th century, a group of Spaniards traveling in the central Peruvian highlands east of modern-day Lima encountered an old Indian carrying khipu that he insisted held a record of “all [the Spanish] had done, both the good and the bad.” Angered, the Spanish burned the man’s khipu, as they did countless others over the years.

Some of the knots did survive, though, and for centuries people wondered if the old man had been speaking the truth. Then, in 1923, an anthropologist named Leland Locke provided an answer: The khipu were files. Each knot represented a different number, arranged in a decimal system, and each bundle likely held census data or summarized the contents of storehouses. Roughly a third of the existing khipu don’t follow the rules Locke identified, but he speculated that these “anomalous” khipu served some ceremonial or other function. The mystery was considered more or less solved.

Then, in the early 1990s, Urton, one of the world’s leading Inca scholars, spotted several details that convinced him the khipu contained much more than tallies of llama sales. For example, some knots are tied right over left, others left over right. Urton came to think that this information must signal something. Could the knotted strings also be a form of writing? In 2003, Urton wrote a book outlining his theory, and in 2005 he published a paper in Science that showed how even khipu that follow Locke’s rules could include place-names as well as numbers.

Urton knew that these findings were a tiny part of cracking the code and that he needed the help of people with different skills. So, early last year, he and a graduate student, Carrie Brezine, unveiled a computerized khipu database – a vast electronic repository that describes every knot on some 300 khipu in intricate detail. Then Urton and Brezine brought in outside researchers who knew little about anthropology but a lot about mathematics. Led by Belgian cryptographer Jean-Jacques Quisquater, they are now trying to shake meaning from the knots with a variety of pattern-finding algorithms, one based on a tool used to analyze long strings of DNA, the other similar to Google’s PageRank algorithm. They’ve already identified thousands of repeated knot sequences that suggest words or phrases.

Now the team is closing in on what might be a writing system so unusual that it remained hidden for centuries in plain sight. If successful, the effort will rank with the deciphering of Egyptian hieroglyphics and will let Urton’s team rewrite history. But how do you decipher something when it looks completely unlike any known written language – when you’re not even sure it has meaning at all?

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URTON WORKS A FEW MINUTES’ WALK from Harvard Yard, in a redbrick building with dark wooden doors and copper gutters that also serves as the university’s Museum of Natural History. But his fifth-floor office is more Lima than Cambridge. Behind his modest desk hangs a Peruvian pan flute. Spanish-language posters adorn the walls. The space is awash in earthy browns – straw-colored carpet, a darker shade for the faux-clay clock face – offset by colorful weavings hung from every wall. Each object is a memento from his many trips to South America to track down khipu.

Today at least 750 khipu survive, scattered about in museums and private collections. Each one has a long primary cord, typically about a quarter-inch in diameter, from which hang smaller “pendant” cords – sometimes just a couple, sometimes many hundred. The pendant cords are tied in a series of neat, small knots. Originally dyed in rich colors, the average khipu has now faded so much it resembles a dirty brown mop head.

How could the Inca have used strings to write? In a sense, any written text is just a record of physical actions. You put a pen to paper and then choose from a prescribed set of options how to move and when to lift up. Each decision is preserved in ink. The same can be done with string. The writer makes a series of decisions, recorded as a knot that can then be read by anyone who knows the rules.

Back in the ’20s, Locke began with the observation that the Inca tied their khipu with three types of knots. There is a “figure-eight” knot, which represents one of something. There are “long” knots, with two to nine turns, representing those numbers. And there are “single” knots, which represent tens, hundreds, thousands, or ten thousands, depending on where they fall on the string. When a khipu is placed flat on the ground, the bottom row is the ones place and successively higher rows stand for higher places. So, the number 327 would have three single knots in the hundreds place. A little lower would be two single knots. Lower still would be a long knot with seven turns.

Most anthropologists assumed that was all there was to it – until 1992. That’s when Urton spent a day looking at khipu in the American Museum of Natural History in New York with his friend Bill Conklin, an architect and textile expert. As he studied the cords, Conklin had an isn’t-that-funny insight: The knots that connect the small pendant strings to the primary cord are always tied the same way, but sometimes they face forward and sometimes backward. Startled, Urton soon noticed additional construction details – such as whether a fiber had been dyed to have a bluish or a reddish tint. All told, Urton has found seven additional bits of binary information that might signal something. Perhaps one means “read this as a word, not a number.” Perhaps the binary code served as a kind of markup language, allowing the Inca to make notes on top of Locke’s number-recording system. And perhaps the 200 or so anomalous khipu don’t follow Locke’s rules because they’ve transcended them.

Most Incan scholars are intrigued by Urton’s ideas, though a few skeptics have noted that he has not produced any proof that his binary code carries meaning, much less that the khipu contain narratives. The Harvard professor concedes that some of the information he’s looking at may not signal anything. But he is convinced the khipu have stories to tell, and he has some history on his side. José de Acosta, a Jesuit missionary sometimes called the Pliny of the New World, wrote a description of the khipu at the end of the 16th century. In it, he describes how the “woven reckonings” were used to record financial transactions involving hens, eggs, and hay. But he also noted that the native people considered the khipu to be “witnesses and authentic writing.” “I saw a bundle of these strings,” he wrote, “on which a woman had brought a written confession of her whole life and used it to confess just as I would have done with words written on paper.”

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EGYPTIAN HIEROGLYPHICS, LINEAR B, ancient Mayan writing – all of the great decipherments have been accomplished by a combination of logic and intuition, persistence and flexibility. Decoding scripts is not like looking for a combination that will open a lock. It’s more like rock climbing: You find a foothold, push up, and hope another presents itself.

Jean-Jacques Quisquater – a tall man with a thin crown of wispy white hair – would like to join the pantheon of puzzle solvers. Quisquater directs a large cryptography laboratory at Belgium’s historic Catholic University of Louvain, where he is known for his work on securing smartcards. In the fall of 2003, he came to MIT for a yearlong academic sabbatical. At the time, he had been thinking nostalgically of a trip to Greece 40 years before when he saw the famous undeciphered Phaistos Disc, a small red-brown disc from deep in the second millennium BC covered on either side with a spiral of glyphs – a fish, a shield, an olive branch. Quisquater hoped to find something equally romantic and challenging to work on.

When he heard about the mystery of the khipu, he was immediately enthralled. He soon met Urton, and they teamed up with a father-son pair of MIT computer scientists, Martin and Erik Demaine. The group began chatting, with the mathematicians offering detailed plans about how to sort the data.

The team agreed that one of Quisquater’s graduate students, Vincent Castus, would first try an analysis known as a suffix tree. The method uses a computer to identify all the blocks of characters in a text that repeat themselves. Thus, the word Mississippi would yield several repeated blocks, including issi, iss, and ss. Suffix trees are used in genetic analysis to find the shortest unique pattern in a sample of DNA.

With the khipu database loaded onto his iMac, Castus worked to build a suffix tree from the knots, leaving aside the more complicated binary data on this first pass. He began in May 2006. By October he had worked out all the details and found an astonishing number of repeats: 3,000 different groups of repeated five-knot sequences. Shorter patterns appeared even more often. He found several pairs of khipu linked by large numbers of matches, suggesting that they could be related.

None of this tells us whether the khipu contain words or stories. It’s possible the researchers have found khipu that just happen to include repeated number sequences that are not interesting for any particular reason, or that some khipu are deliberate copies of others.

But Urton suspects there’s more to it than that. He knows repetition is the code-breaker’s great friend. A Cold War sleuth noticing an oft-used sequence might guess it stood for Moscow or Khrushchev. Recognizing repeated place-names was one of the first steps in deciphering the ancient Mycenaean script Linear B. Now the team has a key for all the khipu in the database, allowing them to instantly identify whenever a particular sequence appears. They also have a list of common short sequences – the most obvious candidates for words.

The team had previously made one breakthrough in identifying connections between knots, thanks to Brezine, who has a background in mathematics and just happens to be a weaver on the side. The master of the khipu database, she wanted to find examples of strings with numbers that added up to sums on another khipu. So she developed a simple algorithm and combed through the data.

Her efforts identified a handful of interlinked khipu that had been uncovered together in a cache in Puruchuco, an archaeological site near Lima. The khipu looked like records kept by three successively higher levels of Incan administrators. Add the numbers on one khipu and the sum is found on another, with that sum in turn found on a third. Imagine, for example, that they depict the results of a census. The village counts up its people and then forwards the total to the district. The district records the numbers from several villages and then forwards the results up to the provincial head. Urton and Brezine do not know what is being counted (people? llamas?), but their 2005 Science paper showed for the first time that information flowed between the khipu.

They have also identified what may be the first word. The two higher-level khipu in the census example use an introductory sequence of three figure-eight knots (1-1-1) that does not appear on what they assume is the village-level khipu. Perhaps only the upper layers have the sequence because it is a label for a particular place, used when compiling information from many locations. Maybe, they suggest, the first symbol to be read off a khipu means this: Puruchuco.

Quisquater’s team, meanwhile, is now working on another, even more ambitious way of extracting clues. It depends on thinking of each knot as a node and each khipu as a network and the links being lengths of string.

One of the surprises from the burgeoning new field of network theory is that the role of a particular node can be summarized – in a deep and meaningful way – by a single number. A good example of this is Google’s PageRank algorithm. The power of the company’s search engine comes from its ability to rank Web pages by relevance. On the Web, a link runs from one page to another, like an arrow. The algorithm interprets that as the first page voting for the second one. Votes flow from across the Internet, like streams joining rivers, eventually pooling at the eBays of the world.

The analysis that the team plans for these khipu networks doesn’t exactly mimic PageRank. After all, the string links between knots aren’t unidirectional like arrows; one knot doesn’t point to another. But the concept is the same: If you think about a big mass of information as a network, and analyze it as a network, looking for the thousands of small and big ways that different piles of information relate to one another, you can see things that you wouldn’t notice otherwise.

Vincent Blondel, a Belgian mathematics professor who is a friend of Quisquater’s, recently helped work out the math behind an approach that allows a computer to calculate degrees of similarity between nodes in two separate networks. Like PageRank, the procedure uses voting, but it assigns each node many scores instead of one and employs a more complex scheme for calculating the totals. Type “baseball” into Google and its spiders will race over the Internet, look at links, and spit back that yankees.com is the 11th most useful site for you and seattlemariners.com is the 22nd. If Quisquater’s algorithm were used on the Web, it would return a slew of numbers, some of which would show similarities between different nodes – or knots. So you’d see that the Yankees and Mariners sites are similar because both receive feeds from majorleaguebaseball.com and have outgoing links to the homepages for 29 teams.

When Quisquater’s algorithm is used on khipu, it will reveal knots or groups of knots that always play a certain role in relationship to others. These might be labels or formatting signs. For example, it may turn out that some of the khipu start with sets of knots that say something like “read this as a calendar.” Or collections of khipu may have similar networks of closely related knots, perhaps signaling that they originate from the same geographic area. Or it could even turn out that the anomalous khipu will all have some pattern that signifies “read this as a story.” The results from this technique should come in sometime later this year, and they will provide valuable clues, even if they don’t immediately crack the Inca paradox.

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URTON’S GREAT INSIGHT has been to treat the khipu not just as a textile or a simple abacus but as an advanced, alien technology. Sitting on a poncho draped over the couch in his office, Urton describes a formative trip to a remote Bolivian village where he worked with traditional weavers. Observing these women spin and ply yarn into multicolored tapestries with elaborate symmetries, he caught a glimpse of the Incan mind at work. For an expert weaver, fabric is a record of many choices, a dance of twists, turns, and pulls that leads to the final product. They would have seen a fabric – be it cloth or knotted strings – a bit like a chess master views a game in progress. Yes, they see a pattern of pieces on a board, but they also have a feel for the moves that led there. “You can see inside of it,” Urton says.

It would be all too easy to dismiss the khipu as the work of a less advanced civilization, one that didn’t develop guns, iron, or wheels. But for more than a decade, Urton has assumed that the khipu are evidence of Incan sophistication in ways we have still not grasped.

Acosta, the 16th-century Jesuit missionary, believed this. He traveled throughout the Americas and recorded several observations of khipu in use. He described religious converts memorizing prayers using khipu-like devices made of small stones or kernels of corn. He also described people in a churchyard completing difficult calculations “without making the slightest error … Whoever wants may judge whether this is clever or if these people are brutish,” he wrote, “but I judge it is certain that, in that which they here apply themselves, they get the better of us.”

Gareth Cook is a science reporter at The Boston Globe. He won a 2005 Pulitzer Prize for his reporting on stem cells.